Differential Geometry and Continuum Mechanics

Differential Geometry and Continuum Mechanics
Title Differential Geometry and Continuum Mechanics PDF eBook
Author Gui-Qiang G. Chen
Publisher Springer
Pages 384
Release 2015-08-11
Genre Mathematics
ISBN 331918573X

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This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.

Geometric Continuum Mechanics

Geometric Continuum Mechanics
Title Geometric Continuum Mechanics PDF eBook
Author Reuven Segev
Publisher Springer Nature
Pages 416
Release 2020-05-13
Genre Mathematics
ISBN 3030426831

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This contributed volume explores the applications of various topics in modern differential geometry to the foundations of continuum mechanics. In particular, the contributors use notions from areas such as global analysis, algebraic topology, and geometric measure theory. Chapter authors are experts in their respective areas, and provide important insights from the most recent research. Organized into two parts, the book first covers kinematics, forces, and stress theory, and then addresses defects, uniformity, and homogeneity. Specific topics covered include: Global stress and hyper-stress theories Applications of de Rham currents to singular dislocations Manifolds of mappings for continuum mechanics Kinematics of defects in solid crystals Geometric Continuum Mechanics will appeal to graduate students and researchers in the fields of mechanics, physics, and engineering who seek a more rigorous mathematical understanding of the area. Mathematicians interested in applications of analysis and geometry will also find the topics covered here of interest.

The Geometrical Language of Continuum Mechanics

The Geometrical Language of Continuum Mechanics
Title The Geometrical Language of Continuum Mechanics PDF eBook
Author Marcelo Epstein
Publisher Cambridge University Press
Pages 325
Release 2010-07-26
Genre Science
ISBN 113949046X

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Epstein presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics. Divided into three parts of roughly equal length, the book opens with a motivational chapter to impress upon the reader that differential geometry is indeed the natural language of continuum mechanics or, better still, that the latter is a prime example of the application and materialisation of the former. In the second part, the fundamental notions of differential geometry are presented with rigor using a writing style that is as informal as possible. Differentiable manifolds, tangent bundles, exterior derivatives, Lie derivatives, and Lie groups are illustrated in terms of their mechanical interpretations. The third part includes the theory of fiber bundles, G-structures, and groupoids, which are applicable to bodies with internal structure and to the description of material inhomogeneity. The abstract notions of differential geometry are thus illuminated by practical and intuitively meaningful engineering applications.

Geometry of Incompatible Deformations

Geometry of Incompatible Deformations
Title Geometry of Incompatible Deformations PDF eBook
Author
Publisher Walter de Gruyter GmbH & Co KG
Pages 410
Release 2019-03-04
Genre Science
ISBN 3110563215

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No detailed description available for "Geometry of Incompatible Deformations".

Material Geometry: Groupoids In Continuum Mechanics

Material Geometry: Groupoids In Continuum Mechanics
Title Material Geometry: Groupoids In Continuum Mechanics PDF eBook
Author Manuel De Leon
Publisher World Scientific
Pages 226
Release 2021-04-23
Genre Mathematics
ISBN 9811232563

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This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials.The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline.

Tensor Algebra and Tensor Analysis for Engineers

Tensor Algebra and Tensor Analysis for Engineers
Title Tensor Algebra and Tensor Analysis for Engineers PDF eBook
Author Mikhail Itskov
Publisher Springer Science & Business Media
Pages 253
Release 2009-04-30
Genre Technology & Engineering
ISBN 3540939075

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There is a large gap between engineering courses in tensor algebra on one hand, and the treatment of linear transformations within classical linear algebra on the other. This book addresses primarily engineering students with some initial knowledge of matrix algebra. Thereby, mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises provided in the book are accompanied by solutions enabling autonomous study. The last chapters deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and might therefore be of high interest for PhD-students and scientists working in this area.

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers
Title Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers PDF eBook
Author Hung Nguyen-Schäfer
Publisher Springer
Pages 389
Release 2016-08-16
Genre Technology & Engineering
ISBN 3662484978

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This book presents tensors and differential geometry in a comprehensive and approachable manner, providing a bridge from the place where physics and engineering mathematics end, and the place where tensor analysis begins. Among the topics examined are tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. The book includes numerous examples with solutions and concrete calculations, which guide readers through these complex topics step by step. Mindful of the practical needs of engineers and physicists, book favors simplicity over a more rigorous, formal approach. The book shows readers how to work with tensors and differential geometry and how to apply them to modeling the physical and engineering world. The authors provide chapter-length treatment of topics at the intersection of advanced mathematics, and physics and engineering: • General Basis and Bra-Ket Notation • Tensor Analysis • Elementary Differential Geometry • Differential Forms • Applications of Tensors and Differential Geometry • Tensors and Bra-Ket Notation in Quantum Mechanics The text reviews methods and applications in computational fluid dynamics; continuum mechanics; electrodynamics in special relativity; cosmology in the Minkowski four-dimensional space time; and relativistic and non-relativistic quantum mechanics. Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers benefits research scientists and practicing engineers in a variety of fields, who use tensor analysis and differential geometry in the context of applied physics, and electrical and mechanical engineering. It will also interest graduate students in applied physics and engineering.